Cosmic Microwave Background anisotropies from second order gravitational perturbations

TL;DR

The paper addresses second-order gravitational perturbations on CMB anisotropies, including scalars, vectors, and tensors, within a flat FRW background. It applies Pyne and Carroll's framework to propagate photons along perturbed null geodesics and to express the observed temperature fluctuations $T_O$ in terms of $\delta T^{(1)}$ and $\delta T^{(2)}$, deriving general expressions for these perturbations that incorporate intrinsic, Doppler, gravitational redshift, ISW, and Rees–Sciama effects, valid in the Poisson gauge. By transforming second-order metric perturbations from the synchronous to the Poisson gauge, the authors obtain explicit expressions for $\psi_P^{(1)}$, $\phi_P^{(1)}$, $\chi_{P ij}^{(1)}$, and the second-order quantities $\psi_P^{(2)}$, $\phi_P^{(2)}$, $z_P^{(2)i}$, $\chi_{P ij}^{(2)}$, showing that vector/tensor contributions induced by scalars are subdominant. The analysis concludes that the leading additional effect at second order is gravitational lensing by density perturbations, with potential GW-lensing corrections, while other second-order terms are comparatively small; these results help quantify the level of second-order corrections relevant for precision CMB parameter estimation and potential detectability with future observations.

Abstract

This paper presents a complete analysis of the effects of second order gravitational perturbations on Cosmic Microwave Background anisotropies, taking explicitly into account scalar, vector and tensor modes. We also consider the second order perturbations of the metric itself obtaining them, for a universe dominated by a collision-less fluid, in the Poisson gauge, by transforming the known results in the synchronous gauge. We discuss the resulting second order anisotropies in the Poisson gauge, and analyse the possible relevance of the different terms. We expect that, in the simplest scenarios for structure formation, the main effect comes from the gravitational lensing by scalar perturbations, that is known to give a few percent contribution to the anisotropies at small angular scales.